# Quick Calculus Question



## Superbird (Feb 21, 2014)

For example, ∑((n * x^n) / n^2), n=1 to infinity. I need to find the radius of convergence for x, and the bounds of that convergence.

I can use ratio test to figure out the answer to this problem:
= ((n+1) * x^(n+1) * n^2) / ((n+1)^2 * x^n * n)
= x * n/(n+1).
—> |x * n/n+1| < 1 
—> -1 < x < 1 

Thus the radius of convergence is 1, and the bounds are (-1,1).

...or are they. This is what I have trouble with -- do I use square brackets or round brackets, and how do I figure out which for each bound? I know it has something to do with figuring out whether the series converges for each bound of x, but does that necessarily mean I have to go through the trouble of figuring that out? For instance, would I now have to evaluate 
 ∑((n * (-1)^n) / n^2) and ∑((n * 1^n) / n^2)
again?

(ok, I know it's a bad example but it's an example of the problem I want to solve.) I know that the first of those, with x = -1, is convergent because of the alternating series test, whereas the second, with x = 1, is divergent because p-series. So the bounds would be [-1,1). But do I have to go through that much effort with _every single problem_, or is there an easier way?


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## Murkrow (Feb 21, 2014)

I wouldn't say it's much effort. There's only two bounds to check after all.
(It's _much_ harder in complex analysis when there are infinitely many numbers lying on the boundary)

But no I can't think of faster ways to do it, I'm afraid. I may be wrong though. My advice would be to assume that it's round brackets until proven otherwise, since it's definitely true that it converges within those bounds.


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